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A351543
Even numbers k such that there is an odd prime p that divides sigma(k), but valuation(k, p) differs from valuation(sigma(k), p), and p does not divide A003961(k), where A003961 is fully multiplicative with a(p) = nextprime(p), and sigma is the sum of divisors function.
6
4, 8, 12, 16, 18, 26, 32, 36, 38, 44, 48, 50, 52, 56, 58, 64, 68, 72, 74, 76, 78, 80, 82, 86, 88, 90, 92, 96, 98, 100, 104, 108, 112, 116, 118, 122, 124, 126, 128, 132, 134, 136, 144, 146, 148, 150, 152, 156, 158, 162, 164, 166, 172, 176, 178, 180, 184, 188, 192, 194, 196, 200, 202, 204, 206, 208, 212, 218, 222, 226
OFFSET
1,1
COMMENTS
Even numbers k such that sigma(k) has an odd prime factor prime(i), but prime(i-1) is not a factor of k, and A286561(k, prime(i)) <> A286561(sigma(k), prime(i)). This differs from the definition of A351542 in that prime(i) is not here required to be a factor of k itself. The condition implies also that if there is any such odd prime factor prime(i) of sigma(k), it must be >= 5.
Even numbers k for which A351555(k) > 0.
Question: Is A351538 subsequence of this sequence?
EXAMPLE
12 = 2^2 * 3 is present as sigma(12) = 28 = 2^2 * 7, whose prime factorization contains an odd prime 7 such that neither it nor the immediately previous prime, which is 5, divide 12 itself.
196 = 2^2 * 7^2 is present as sigma(196) = 399 = 3^1 * 7^1 * 19^1, which thus has a shared prime factor 7 with 196, but occurring with smaller exponent, and with no prime 5 (which is the previous prime before 7) present in the prime factorization of 196.
364 = 2^2 * 7^1 * 13^1 is present as sigma(364) = 784 = 2^4 * 7^2, which thus has a shared prime factor 7 with 364, but occurring with larger exponent, and with no prime 5 (which is the previous prime before 7) present in the prime factorization of 364.
PROG
(PARI)
A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A351555(n) = { my(s=sigma(n), f=factor(s), u=A003961(n)); sum(k=1, #f~, if((f[k, 1]%2) && 0!=(u%f[k, 1]), (valuation(n, f[k, 1])!=f[k, 2]), 0)); };
isA351543(n) = (!(n%2) && A351555(n)>0);
CROSSREFS
Subsequences: A351541, A351542, and also conjecturally A351538.
Cf. A351553 (complement among even numbers).
No common terms with A349745.
Sequence in context: A311120 A373729 A311121 * A354515 A277015 A311122
KEYWORD
nonn
AUTHOR
Antti Karttunen, Feb 16 2022
STATUS
approved